Adaptive B-spline volume representation of measured BRDF data for photorealistic rendering

Author's Accepted Manuscript

Adaptive B-spline Volume Representation of Measured BRDF Data for Photorealistic Rendering Hyungjun Park, Joo-Haeng Lee

www.elsevier.com/locate/jcde

PII: DOI: Reference:

S2288-4300(14)00002-5 http://dx.doi.org/10.1016/j.jcde.2014.11.001 JCDE1

To appear in:

Journal of Computational Design and Engineering

Received date: Revised date: Accepted date:

8 November 2014 26 November 2014 27 November 2014

Cite this article as: Hyungjun Park, Joo-Haeng Lee, Adaptive B-spline Volume Representation of Measured BRDF Data for Photorealistic Rendering, Journal of Computational Design and Engineering, http://dx.doi.org/10.1016/j.jcde.2014.11.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive B-spline Volume Representation of Measured BRDF Data for Photorealistic Rendering Hyungjun Park1* and Joo-Haeng Lee2

1 Department of Industrial Engineering, Chosun University, Gwangju 501-759, South Korea 2 Electronics and Telecommunications Research Institute, Daejeon 305-350, South Korea Corresponding author. Tel.: +82-62-230-7039, Fax.: +82-62-230-7128 E-mail address: [email protected] *

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Measured bidirectional reflectance distribution function (BRDF) data have been used to represent complex interaction between lights and surface materials for photorealistic rendering. However, their massive size makes it hard to adopt them in practical rendering applications. In this paper, we propose an adaptive method for B-spline volume representation of measured BRDF data. It basically performs approximate B-spline volume lofting, which decomposes the problem into three sub-problems of multiple B-spline curve fitting along u-, v-, and w-parametric directions. Especially, it makes the efficient use of knots in the multiple B-spline curve fitting and thereby accomplishes adaptive knot placement along each parametric direction of a resulting B-spline volume. The proposed method is quite useful to realize efficient data reduction while smoothing out the noises and keeping the overall features of BRDF data well. By applying the Bspline volume models of real materials for rendering, we show that the B-spline volume models are effective in preserving the features of material appearance and are suitable for representing BRDF data. Keywords: B-spline volume; Measured BRDF; Approximate volume lofting; Adaptive knot placement; Data reduction

1. Introduction Bidirectional reflectance distribution function (BRDF) is a fundamental radiometric concept that models directional properties of reflection between light and surface materials [1,2]. In computer graphics, BRDF has been widely adopted for realistic representation of material appearance. Although there have been extensive works on modeling surface reflectance, it is still an active research topic to develop BRDF models. Since the first introduction of Phong model [3], various BRDF models have been proposed either to overcome the shortcomings of the others or to achieve the different goals such as accuracy, computational efficiency, controllability, versatility, and compactness in data size. The good description of BRDF models can be found in [1,2,3,4,5]. Physically-based rendering aims to simulate the interaction of light and materials by more precisely considering various physical properties of optics to enhance the realism of synthetic images [1,2]. Recently, there have been strong requirements to simulate sophisticated material properties with great accuracy that cannot be easily achieved with existing modeling techniques of material appearance. Advances in measuring technology have made it possible to accurately acquire reflection properties of real materials using a gonioreflectometer [6,7,8]. Obviously, using measured BRDF data has been one of the alternatives to simple phenomenological models. For example, measured BRDF data are actively used for virtual prototyping in fashion, consumer electronics, and automotive industries to visualize the sophisticated real materials [9,10], and for VFX in the film industry to express delicate textile materials of digital doubles [11]. To be applied in such applications, the measured BRDF data should be stored into a simple tabular form or fit into a more sophisticated representation. As the measured BRDF data are usually huge and even noisy, it is much preferable to compute suitable mathematical models which approximate the data and smooth out the noise in the data. When the BRDF data is used in the tabular form without compression, their size causes a critical problem in production environment. For example, in case of the complete BRDF data set of a single material, its storage size varies from 33 to 750 megabytes, which can be a serious burden even to a commercial rendering system handling multiple sets of such BRDF data along with several gigabytes of geometry and texture

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

data, not only in a memory but also over the network. Thus, it is very important to develop a compact representation scheme that preserves the overall features of the data faithfully.

A general approach is to approximate the measured BRDF data by one of the analytical models with a small set of control parameters [12]. However, as most of the analytical models are defined by nonlinear functions, it is required to employ nonlinear optimization algorithms for estimating the underlying parameters. Obviously this nonlinear optimization approach has several shortcomings. According to Ngan et al. [12], most of the analytical models obtained by non-linear least squares fitting could not express all the reflection features such as generalized diffuse, backscattering, off-specular, and Fresnel effects. For better results, it is required to increase the number of lobes or parameter complexity, which usually increase not only the difficulty in solving the non-linear optimization problem but also the computational burden of evaluating BRDF values from the fitted models. In an effort to reduce the size of measured BRDF data while keeping an accurate representation of the data, many alternative approaches have been proposed. Shröder and Sweldens [13] used spherical wavelets to represent reflectance functions. Several research groups used polynomial based analytical models. Koenderink et al. [14] used Zernike polynomials for representing BRDF data, and Ozturk et al. [15] represented diffuse and glossy BRDF data by polynomial functions combined with the principal component analysis. In most cases of using polynomial based analytical models, a higher degree of polynomials is required to accurately fit some complex data. Several research groups including Lawrence et al. [16] have investigated techniques for factoring large BRDF datasets into more compact, manageable form. In all cases, the four dimensional (4D) anisotropic BRDF is factored into products of two dimensional (2D) functions that can be represented as texture maps. In most cases, the factorizations are not so accurate as the original measured BRDF data. Recently, Kurt and Cinsdikici [17] presented a new BRDF model which uses neural networks SOMs and MANs to represent measured BRDF data. B-splines have been one of de facto industrial standards to represent curves and surfaces in the field of Computer Aided Geometric Design (CAGD) for more than a decade [18,19]. It is well-known that higher-degree polynomial models are inefficient to process and are numerically unstable to be fitted [18,19]. As B-splines are defined by piecewise polynomials, they can overcome the shortcomings of polynomial based models. Although many useful methods have been presented for B-spline curve or surface fitting to measured data, comparatively little attention has been given to how to apply B-spline volume fitting for rendering. Lee and Park [20,21] proposed a method for B-spline volume representation of isotropic measured BRDF data. Their method works well when the values of the given data changes smoothly (i.e., for BRDF data of materials which are diffuse or moderately glossy), but it has difficulty when the data values change rapidly or have sharp peaks considered as not outliers but features (i.e., for BRDF data of glossy or specular materials). In subsequent work in the form of short note [22], they coined the idea of adaptive placement of knots of resulting B-spline volumes as a way of alleviating the shortcoming. In this paper, we solidify the idea coined in [22] and propose an adaptive method for B-spline volume representation of measured BRDF data. The method basically performs approximate B-spline volume lofting, which decomposes the problem into three sub-problems of multiple B-spline curve fitting along u-, v-, and w- parametric directions. Especially, it makes the efficient use of knots in the multiple B-spline curve fitting and accomplishes adaptive knot placement using dominant columns along each parametric direction of a B-spline volume. The resulting B-spline volume called B-spline volume BRDF (hereafter BVB) becomes a compact but accurate representation of the measured BRDF data. We show that the proposed method is quite useful to realize efficient data reduction while smoothing out the noises and keeping the overall features of BRDF data. By applying the BVB models of real materials for rendering, we also show that the BVB models are effective in preserving the features of material appearance and are suitable for representing the measured BRDF data coming from a wide range of materials (i.e., not specific to diffuse and moderately glossy materials). The rest of the paper is organized as follows. In section 2, we explain some preliminaries related to BRDF. In section 3, we present the adaptive method for BVB model fitting. In Section 4, we describe the application of BVB models to photorealistic rendering. In Section 5, we explain experimental results to show that BVB models are appropriate for BRDF representation and are useful to enhance rendering process. Finally, we conclude the paper with some remarks in Section 6.

2. Preliminaries Related to BRDF 2.1 Measured BRDF Data For two unit vectors ω i and ω o denoting input (or incident) and output (or reflected) directions of light rays at a surface point p, BRDF is the differential ratio of reflected radiance along ω o to the irradiance incident on the point p along ω i [1,2]. At a point p in the spherical coordinate system, BRDF is defined as f (ω o ,ωi ) = f (θ o ,φo ,θ i , φi ) where ωi = (θ i ,φi )

and ω o = (θ o , φo ) . If BRDF can be defined as f (θ o ,θ i , φd ) where φ d = φo − φi , such BRDF is called isotropic. Otherwise, it is called anisotropic. As shown in Figure 1, the domain of isotropic BRDF is 3D rather than 4D. Examples of anisotropic materials include brushed metals and velvets. Most of ordinary materials are isotropic. In this paper, we deal with only isotropic BRDF.

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

Figure 1. Illustration of isotropic BRDF. Image-based measurement equipment is actively used due to its relatively simple configuration [6,7,8]. While a special light source rotates along the spherical sample of a target material, a digital camera captures the images of the sample at each light position. After post-processing, we can obtain an isotropic BRDF data set which is stored in a simple tabular form (hereafter TBF): fi , j , k = f (θi = ai ,θo = b j ,φd = ck ) (1) where (i , j , k ) ∈ [0,.., imax ] × [0,.., jmax ] × [0,.., k max ] . The angle values a i , b j , ck are defined as follows: ai = π2 i imax , bj =

π 2

j jmax , ck = 2π k k max . When RGB triple is used, each BRDF data element should be a vector of three color coordi-

nates as follows: fi , j , k = ( fi ,r j , k , fi ,gj , k , fi ,bj , k ) . Thus, the measured BRDF data are given as follows: F = {fi , j ,k | i ∈ [0,.., imax ], j ∈ [0,.., jmax ], k ∈ [0,.., kmax ]} .

(2)

Figure 2 shows the iso-meshes of measured BRDF data sets obtained from various materials. In the figure, the iso-meshes are displayed at specific incident angles in the spherical coordinate system.

Figure 2. Iso-meshes of measured BRDF data obtained from various materials. When each color coordinate is encoded as four bytes and the measurement step is 1 degree for each dimension (i.e., imax = jmax = 90 , k max = 360 ), the size of measured BRDF data of each material reaches to 70 megabytes. This is a very large

size for both off-line and real-time renderers. Hence, we are interested in developing a new representation scheme that requires small storage while preserving the reflection features of the original material. 2.2 Fitting Analytical Models to Measured BRDF data Ngan et al. [12] evaluated several analytical models in terms of their ability of approximating measured BRDF data. Due to practical problems of non-linear fitting (such as dependency on initial guesses and convergence to local optima) and limited power of expression, it is difficult to find proper parameter values of each analytical model that preserves all the overall features of the BRDF data. Thus the fitted models often do not fit the data satisfactorily. As the Lafortune model (hereafter LFT) [4] and the Cook-Torrance model (hereafter CKT) [5] have been widely used among analytical models, we use them as reference models to assess the usefulness of BVB models for data fitting and rendering. According to Ngan et al. [12], the CKT model outperformed the other analytical models, and the LFT model showed the average performance for fitting and rendering. The CKT model, which originated from the field of physics, has been adopted for computer graphics, specifically to support more physically based properties of light reflection on the surface geometry [5]. The LFT model is a generalization of the Phong model, which is expressed as a sum of multiple specular lobes [4]. It is useful in representing relatively simple reflection features with flexibility of control, and it is efficient in evaluation. However, it is not suitable to represent materials with complex reflection features. Moreover, it fits the data poorly in the region of grazing angles where the light vector is nearly parallel to the tangent plane at each surface point.

0000

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

3. Adaptive Method for BVB Model Fitting 3.1 B-spline Volume A parametric B-spline curve of order p is defined as follows: n

C ( t ) = ∑ i = 0 N i , p (t ) b i

where

bi

are control points,

(3)

are the normalized B-spline functions defined over the knot vector

N i , p (t )

T = {t0 , t1 ,..., tn + p −1 , tn + p } , and the parameter domain is given as t ∈ [t p −1 , tn +1 ] . By directly extending B-spline curves to a triva-

riate case, we define a parametric tensor product B-spline volume B(u, v, w) of order (degree+1) p, q, r as follows: nu

nv

nw

B (u , v , w) = ∑ ∑ ∑ N i , p (u ) N j , q ( v ) N k , r ( w)b i, j , k

(4)

i=0 j =0 k =0

where bi , j , k are control points, and the parameter domains are given as u ∈ [u p −1 , unu +1 ] , v ∈ [vq −1 , vnv +1 ] , and w ∈ [wr −1 , wnw +1 ] . N i , p (u ) , N j , q (v) , and Nk , r ( w) are the normalized B-spline functions of order p, q, r defined on knot vec-

tors U, V, W in the u, v, w directions, respectively. The knot vectors U, V, W are given as U = {u0 , u1 ,.., u n

u

{

}

{

+p

},

}

V = v0 , v1 ,.., vnv + q , and W = w0 , w1 ,.., wnw + r . In this work, we pursue a proper way of fitting a B-spline volume to the meas-

ured BRDF data F of a real material given in Equation (2). The resulting B-spline volume becomes a BVB model. 3.2 Approximate B-spline Volume Lofting We want each element fi , j , k of the BRDF data F to be a B-spline volume element at specific parameter values (u, v, w) = (ui , v j , wk ) , that is, B(ui , v j , wk ) = fi , j , k . Hence, the problem of constructing a B-spline volume B(u, v, w) from the

data F can be solved by the least-squares B-spline volume fitting. When the parameter values u i , v j , w k of the volumetric data fi , j , k , the orders p, q, r of B-spline functions, and the knot vectors U, V, W of a B-spline volume B(u, v, w) are given, the control points bi , j , k can be determined by minimizing the following least-squares error: imax jmax k max

E (b 0,0,0 ,..., b nu , nv , nw ) = ∑ ∑ ∑ B (u i , v j , wk ) − f i , j , k

2

.

(5)

i=0 j =0 k =0

According to the uniform angle steps in Equation (1), the parameter values u i , v j , w k are equally spaced as ui = i / imax , v j = j / jmax , and wk = k / kmax , respectively. It is natural to make the parameters values linearly related to the angle values. The knots of the knot vectors U, V, W can be placed using the parameter values [18,19]. Since the objective functional in Equation (5) is quadratic, this minimization problem leads to that of solving a system of linear equations. Without loss of generality, imax > nu , jmax > nv , and k max > nw , so the least-squares minimization provides reasonably good solutions. However, as the size of the system matrix becomes very large (that is, M × M where M = ( nu + 1) × ( nv + 1) × ( nw + 1) ), the direct approach to solving such a large linear system should be avoided due to inefficiency in computation time and memory usage. The proposed method for B-spline volume representation of BRDF data basically takes the scheme of approximate Bspline volume lofting, which is extended from approximate surface lofting based on the property of tensor products of Bsplines [23,24]. In the approximate volume lofting, we construct a B-spline volume by successively applying multiple Bspline curve fitting to a set of polylines along each parametric (w-, v-, u-) direction. Consider that the equations B(ui , v j , wk ) = fi , j , k are written as follows: fi , j , k = B(ui , v j , wk ) n

= ∑ c w= 0 N c , r ( wk )

By replacing

nu

nv

a=0

b =0

∑ ∑

N a , p (u i ) N b , q ( v j )b a,b , c by intermediate control points

(∑

nu a =0

∑

nv b =0

N a , p (ui ) N b, q (v j )b a,b, c

)

(6)

g i , j ,c , we can simplify Equation (6) as fol-

lows: n

f i , j , k = ∑ c w= 0 N c , r ( wk ) g i , j , c = C i , j ( wk )

for (i, j ) ∈[0,.., imax ] × [0,.., jmax ] . (7) Equation (7) means multiple B-spline curve fitting to (imax + 1) × ( jmax + 1) polylines each of which consists of (k max + 1) points passing along the w direction. We can obtain the control points g i , j , c for (i, j , c ) ∈ [0,.., imax ] × [0,.., jmax ] × [0,.., nw ] by solving the multiple B-spline curve fitting where n w , r, wk , and W are given. Now the intermediate control points g i , j , c

0000

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

are expressed as follows: n

g i , j , c = ∑ bv= 0 N b , q (v j )

By replacing

∑

nu a =0

(∑

nu a=0

N a , p (ui )b a,b , c

)

(8)

N a , p (u i )b a,b , c by intermediate control points hi ,b,c , we can simplify Equation (8) as follows: nv

g i , j , c = ∑ N b , q ( v j )h i , b , c b =0

Equation (9) means multiple B-spline curve fitting to (imax points

passing

along

the

v

direction.

Similarly,

for (i, c ) ∈ [0,.., imax ] × [0,.., nw ] . (9) + 1) × ( nw + 1) polylines each of which consists of jmax + 1 we can obtain the control points hi ,b,c for

(i, b, c ) ∈ [0,.., imax ] × [0,.., nv ] × [0,.., nw ] by applying multiple B-spline curve fitting where nv , q, v j , and V are given. The

intermediate control points hi ,b,c are expressed as follows: n

h i , b , c = ∑ au= 0 N a , p (ui )b a,b , c

for (b, c ) ∈ [0,.., nv ] × [0,.., nw ] . (10) Equation (10) means multiple B-spline curve fitting to ( nv + 1) × ( nw + 1) polylines each of which consists of imax + 1 points passing along the u direction. We can finally obtain the control points ba,b,c for (a, b, c) ∈ [0,.., nu ] × [0,.., nv ] × [0,.., nw ] by solving the multiple B-spline curve fitting where nu , p, u i , and U are given. The overall process of approximate B-spline volume lofting is depicted in Figure 3 and its main steps are summarized as follows:

Figure 3. Overall process of approximate B-spline volume lofting. (1) From the input points fi , j , k , construct w-directional polylines, each of which has the same number of points k max + 1 . Determine wk and W along w-direction. Then, do w-directional multiple B-spline curve fitting by Equation (7), which converts input points fi , j , k to intermediate control points g i , j , c for (i, j, c) ∈[0,.., imax ] ×[0,.., jmax ] ×[0,.., nw ] . (2) From the intermediate points g i , j , c , construct v-directional polylines, each of which has the same number of points jmax + 1 . Determine v j and V along v-direction. Then, do v-directional multiple B-spline curve fitting by Equation (9),

which converts the points g i , j , c to intermediate control points hi ,b,c for (i, b, c) ∈[0,.., imax ] ×[0,.., nv ] × [0,.., nw ] . (3) From the intermediate points hi ,b,c , construct u-directional polylines, each of which has the same number of points imax + 1 . Determine u i and U along u-direction. Then, do u-directional multiple B-spline curve fitting by Equation (10),

which converts the points hi ,b,c to final control points ba,b,c for (a, b, c) ∈[0,.., nu ] ×[0,.., nv ] ×[0,.., nw] . (4) Construct a B-spline volume B (u , v, w) with the control points b a,b , c . The control points ba,b,c of the B-spline volume B(u, v, w) are obtained by applying multiple B-spline curve fitting three times, which converts points fi , j , k to intermediate points g i , j , c and hi ,b,c , and finally to ba,b,c .

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

3.3 Multiple B-spline curve fitting with adaptive knot placement The multiple B-spline curve fitting is a key ingredient of approximate B-spline volume lofting. Given a set of polylines, it constructs a set of compatible B-spline curves each of which fits the points of its corresponding polyline [23,24]. B-spline curves are called compatible when they have the same B-spline order, the same number of control points, and the same knot vector. In the multiple B-spline curve fitting derived from Equations (7), (9), or (10), the polylines have the same number of points. Given mr + 1 polylines Pi = {pi , j | j = 0,..., mc } for i ∈ [0,.., mr ] , consider how to construct B-spline curves of order p n

which are defined as C i (t ) = ∑ k = 0 N k , p (t )b i , k . As the polylines have the same number of points, all the points of each column Q j = {pi , j | i = 0,.., mr } are made to have the same parameter value t j , that is, Ci ( t j ) = pi , j for i ∈ [0,.., mr ] . The values t j are equally spaced (i.e. t j = j / mc ) as mentioned in Section 3.2. The knots of a common knot vector T = {t0 , t1 ,.., tn + p } are usually chosen to reflect the distribution of the parameter values [18,19,23,24,25,26]. The control

points b i , k for k ∈[0,.., n] of the ith B-spline curve Ci (t) can be determined by minimizing the following least-squares errors: m

E (b i ,0 ,..., b i , n ) = ∑ j =c 0 C i ( t j ) − p i , j

2

for i ∈ [0,.., m r ] .

(11)

Each minimization problem is equivalent to solving a linear system. Each resulting curve Ci (t) is a piecewise B-spline curve of order (degree+1) p with Cp-2 continuity at interior knots [18,19]. It is easily found that the successive application of multiple B-spline curve fitting using Equation (11) is much more efficient in terms of computation time and memory usage than the direct approach using Equation (5). Recall that m c denote the number of points of each polyline and n the number of control points. The results of multiple curve fitting are affected by how to place the knots in Equation (11). Several ways of placing the interior knots of the knot vector T have been suggested including the averaging technique for mc = n , and the knot placement techniques (hereafter KTP) for mc > n [18,19]. When m c is nearly greater than n ( | mc − n | is small), it often generates undesirable results [23]. To avoid this, Piegl and Tiller [25] suggested another knot placement technique (hereafter NKTP). Note that, in Lee and Park’s work [20,21] on B-spline volume representation of BRDF data, knots are determined by the knot placement techniques KPT and NKTP. However, these knot placement techniques select the knots in a simple and trivial manner that each knot span contains almost the same number of parameter values, and that all interior knots are changed even though the number n increases by one, which makes it difficult to realize adaptive curve fitting and prevents compact and accurate B-spline volume representation of BRDF data when the data values change rapidly or have sharp peaks considered as not outliers but features (i.e., for BRDF data of glossy or specular materials). Recently, Park and Lee [26] presented a novel approach (hereafter DOM) to knot placement for B-spline curve fitting. After selecting dominant points from the given points, we can determine the interior knots by averaging the parameter values of the dominant points. This knot placement results in a stable system matrix which is not singular, and supports local modification realizing adaptive curve fitting that fewer knots are placed at flat regions but more at complex regions. We extend the DOM approach for multiple B-spline curve fitting where a set of polylines are considered instead of a single polyline. Note that the points of each column Q j = {pi , j | i = 0,.., mr } have the same parameter value t j . In multiple Bspline curve fitting to a set of ( mr + 1 ) polylines Pi = {p i , j | j = 0,..., mc } , we select (n+1) dominant columns D k = Q f ( k ) from ( mc + 1 ) columns Q j , determine a common knot vector T

by averaging the parameter values t j associated with

the dominant columns Dk as follows: t p + i −1 =

1 p −1

∑

i+ p−2 k =i

t f ( k ) for i = 1,.., n − p + 1

(12)

where f ( k ) is a monotonically increasing function that returns the index of the column corresponding to the kth dominant column (i.e. Q f ( k ) = Dk ). Then, with these knots, we obtain compatible B-spline curves Ci (t ) by minimizing the leastsquares errors in Equation (11). Figure 4 shows multiple B-spline curve fitting with and without adaptive knot placement.

0000

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

(a)

(b)

(c)

(d)

(e) (f) Figure. 4. Multiple B-spline curve fitting: (a) 11 polylines each of which consists of 151 points; (b) 17 dominant columns drawn in blue with the polylines; (c) compatible B-spline curves obtained using the dominant columns; (d) control polygons each of which consists of 17 control points; (e) compatible B-spline curves obtained by NKTP; (f) control polygons (17 control points per each) obtained by NKTP. To determine the dominants columns, we employ an adaptive refinement approach that places a new dominant column in the subset with the largest deviation at the current iteration. Note that the points p i , j = ( xi , j , yi , j , zi , j ) do not have any geometric meaning in Euclidean space because they come from BRDF data in Equation (2). Thus, during the determination of the dominant columns, we transform the points pi , j into pˆ i , j = ( xˆi , j , yˆ i , j , zˆi , j ) whose coordinates are given as xˆi , j = i mr , yˆ i , j = t j ,

zˆi , j =

2

2

(λ x ) + (λ y ) + (λ z ) 1 i, j

2 i, j

3 i, j

2

.

(13)

In this work, the same weight was set to each coordinate (i.e., λ1 = λ2 = λ3 = 1 ). Now consider a subset of columns S s , e = {Q j | j = s ,.., e} in which there are only two dominant columns Q s and Qe . Starting with an initial subset S 0, mc , we can obtain as many dominant columns as required by repeating the following steps: Among subsets of columns, find a subset S s,e ( | e − s |> 1 ) with the largest deviation, and choose a new dominant column s,w and S w,e . Then, divide the subset Q w ( s < w < e ) which minimizes the difference between the total arc lengths of S

S s,e into two subsets S s , w and S w,e .

The deviation Ds,e of a subset S s,e can be defined in various ways. In this work, we define it as the maximum chord height s,e Dchd_avg =

mr s ,e e ˆ
Dchd _ max = max j = s max i = 0 p i , j − p i , j 1 ( mr +1)( e − s +1)

e

mr

j=s

i =0

∑ ∑

pˆ i , j − p
i , j

or

the

average

chord

heights

where p
i , j is the orthogonal projection of the pˆ i, j onto the line segment

pˆ i , s pˆ i , e (i.e., chord height of the pˆ i, j with respect to the line segment). The total arc length Ls , e of the subset S s,e is

defined as Ls ,e =

mr

e −1

i =0

j=s

∑ ∑

pˆ i , j +1 − pˆ i , j

.

The maximum chord height tends to make dominant columns closer at sharp regions than the average chord height. For the BRDF data of diffuse or moderately glossy materials, as their polylines usually have smooth shape without sharp peaks, the maximum chord height is likely to be good to locate the dominant columns. For the BRDF data of glossy or specular

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

materials, as their polylines usually have sharp peaks at small regions, the average chord height is enough to locate the dominant columns. We have experienced that the maximum chord height is usually good for the BRDF data sets of diffuse or moderately glossy materials, and that the average chord height is good for the BRDF data sets of glossy or specular materials. As shown in Figure 4, the multiple B-spline curve fitting using dominant columns has a notable tendency to generate fewer control points (correspondingly, curve segments) at flat regions but more at complex regions, which is very helpful to approximate BRDF data by a compact and accurate B-spline volume consisting of fewer control points. Note that the points in Figure 4 are the ones transformed from an iso-mesh of the BRDF data.

3.4 Error-bounded BVB model fitting When B-spline order ( p, q, r) , the number of control points ( nu , nv , nw ) , and the measured BRDF data of any real material are given, we can generate a BVB model to represent the BRDF data by applying the adaptive method of BVB model fitting. For practical applications, it is more useful to specify a tolerance in order to obtain a BVB model satisfying that the fitting error between the BVB model and the given BRDF data is smaller than the tolerance. The resulting BVB model is called error-bounded. In order to get a compact but accurate BVB model using the error-bounded BVB model fitting, it is important to reduce the number of knots (correspondingly control points) while keeping desired accuracy. As the problem of BVB model fitting is decomposed into sub-problems of multiple B-spline curve fitting, it can be completed with the successive application of error-bounded multiple B-spline curve fitting along each parametric direction. Thus, the given tolerance is equally divided into three, each for the error-bounded multiple B-spline curve fitting along its corresponding (u-, v-, or w-) parametric direction where the error is defined as E = max im=r0 max mj =c 0 p i , j − Ci ( t j ) . As it is not known in advance how many control points are required for the error-bounded multiple B-spline curve fitting, it is common to take an iterative (incremental or binary search-based) process which repeats fitting, checking deviation, and adjusting the number of knots (i.e., the number of control points). In this work, we take the binary search algorithm [23,24] to complete the process since it is better in computation.

4. Application of BVB Models to Rendering Given the BRDF data of any material, we can now generate a B-spline volume to represent the BRDF data by the proposed method for BVB model fitting. Then, we can use the B-spline volume (i.e. BVB model) as an alternative BRDF model for photorealistic rendering. Figure 5 shows a conceptual framework of photorealistic rendering employed in this work.

Figure 5. Framework of rendering using three types of BRDF models.

It has three shaders for measured BRDF data, analytical BRDF models, and BVB models. They are integrated into PBRT renderer [2], which provides C++ API to extend its basic functionalities. Each shader is invoked with its associated BRDF model at run time when surfaces of interest should be rendered. The PBRT renderer supports several materials based on classic BRDF models such as perfect diffuse, Lafortune (LFT), Cook-Torrance (CKT). In our case, we added new C++ classes that inherit the abstract BRDF classes. For BVB models, we apply the proposed method for B-spline volume representation of the measured BRDF data of 100 materials obtained by Matusik et al. [7,8]. For analytical BRDF models, we use the results of Ngan et al. [12] where they conducted analytical BRDF model fitting to the measured BRDF data of the 100 materials. For photorealistic rendering, we employed distribution ray tracing in PBRT [2]; we used direct lighting with 64 pixel samples and 64 light samples while applying single environmental lighting with a high dynamic range image and setting the maximum trace depth to 4. In order to decouple the effect of importance sampling [16] on the quality of rendering, we applied cosine-weighted uniform hemisphere sampling [2]. All the rendered images were generated based on single threaded processing.

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

Figure 6. Images rendered with BRDF modelss (from left to right, TBF, BVB, LFT, and CKT) of alum bronze. bronze

Figure 7. Images rendered with BRDF modelss (from left to right, TBF, BVB, LFT, and CKT) of green metallic paint. paint

Figure 8.. Images rendered with BRDF models (from left to right, TBF, BVB, LFT, and CKT) of red plastic. Figures 6 to 8 show happy Buddha and sphere images images which were generated by photorealistic rendering using four kinds of BRDF models (TBF, BVB, LFT, and CKT) T) for three materials (alum bronze, green metallic paint, and red plastic). plastic Recall that TBF denotes a tabular lar form of measured BRDF data. In this work, rk, the size of rendered images is set to 256x512 and 512x512 pixels for happy Buddha and sphere scenes, re-

0000

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

spectively. For rendering with BVB models, we generated BVB models by error-bounded BVB model fitting with B-spline orders and tolerance rates specific to their associated BRDF data. Note that the images rendered using BVB models are very faithful to the images rendered using TBF models.

5. Experimental Results We implemented the adaptive BVB model fitting proposed in Section 3 and the photorealistic rendering described in Section 4. The implementation has been completed using C and C++ languages on an IBM compatible personal computer running MS Windows Vista with an Intel Xeon processor X5365. In this section, we show the experimental results of generating BVB models and of applying them to rendering in order to demonstrate that the BVB models are compact and faithful to represent the BRDF data for rendering applications. We also compare the BVB models with two types of analytical BRDF models (CKT and LFT) in accuracy of data fitting and quality of rendering. For test datasets, we use isotropic BRDF data of 100 real materials measured by Matusik et al. [7,8]. The size of each BRDF data is given as (imax , jmax , kmax ) = (90,90,360) . We found that all the BRDF data sets are more or less contaminated with noise and especially the data sets coming from specular materials contain sharp peaks considered as features.

5.1 Generation of BVB models and their application to rendering For given B-spline orders ( p, q, r ) , we applied the proposed method to obtain a BVB model of each material either by specifying the number of control points ( nu , nv , nw ) or by specifying the tolerance. Then, we applied photorealistic rendering using the BVB model. To evaluate the quality of a BVB model, we computed the error between the BVB model and its corresponding BRDF data. In this work, we use the average fitting error AE which is defined as follows: AE = ( i +1)( j 1 +1)( k +1) ∑ B (ui , v j , wk ) − fi , j , k . (14) max

max

max

∀(i , j , k )

As simple statistical values of the BRDF data, we use the average and the largest values (AR, MR) of the BRDF data which are defined as follows: AR = ( i +1)( j 1 +1)( k +1) ∑ fi , j , k , MR = max fi , j , k . (15) max

max

max

∀(i , j , k )

∀(i , j , k )

For the 100 materials used in this work, their MR values (0.4~8907.5) have great variation (std=1594.8), but their AR values (0.008~0.24) have small variation (std=0.056). We used the MR values to roughly classify the materials into three: diffuse, glossy specular, and highly specular materials. Diffuse materials mostly have small MR values (i.e. 0.76 for blue fabric and 3.49 for polyethylene), but highly specular materials have large MR values (i.e. 4533.6 for aluminum and 8907.5 for chrome). Glossy specular materials have in-between MR values (i.e. 31.4 for gold paint and 196.8 for purple paint). To evaluate the quality of a BRDF model in view of rendering, we estimated the difference error between an image rendered using TBF (i.e. target image) and an image rendered using its fitted model (i.e. test image). For more consistent estimation, we made the two images have the same background image in TIFF format. Let ai , j denote a pixel of a target image a and b i , j a pixel of a test image b. Each image size is ( xmax + 1) × ( ymax + 1) , and the RGB value of each image pixel varies from 0 to 255. To assess the test image b with respect to the target image a, we use the mean absolute error (MAE) defined as follows: (16) MAE = ( x +1)1i ( y +1) ∑ a i , j − b i , j . max

max

∀ (i , j )

BVB model fitting by specifying the number of control points is the basic tool for generation of a BVB model of each material. As an example of such BVB model generation, we constructed from the BRDF data of gold metallic paint ( AR = 0.092 , MR = 4.074 ) a set of B-spline volumes. Figure 9 shows some graphical results of a BVB model where ( nu + 1, nv + 1, nw + 1) = (15,15,60) and p = q = r = 4 . It is found that AE = 0.003 and ME = 0.343 for the BVB model. The first row of the figure shows three iso-meshes ( θi = 30 ,60 ,90 ) in a (θ o ,φd ) spherical coordinate system for the TBF model. The second row of the figure shows three corresponding iso-surfaces ( u = 13 , 32 ,1 ) in a (v, w) spherical coordinate system for the BVB model. Note the BVB model can smooth out the noises and keep the overall feature of the data. Whereas the storage size of the BRDF data reaches to 70 megabytes in TBF (35 megabytes in halfway direction representation [7]), the size of the BVB model is about 320 kilobytes (0.46 % of the input data).

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

Figure 9. Iso-meshes of a TBF model and a BVB model at three incident angles. (From left to right, θl = 30 ,60 ,90 ) Figure 10 shows happy appy Buddha images rendered using four kinds of BRDF models for gold metallic paint. Figure 10(a) is a target image obtained using the TBF model. model Figures 10(b) to (d) are test images obtained using BVB,, LFT, and CKT models. The BVB model is the same one used in Figure 9. 9 Figures 10(e) to (g) are the difference images between the target imi age and the test images. In case of the difference image imag for the BVB model in Figure 10(e), difference ference values are scaled up 20 times for easy recognition. The difference image errors (MAE) ( are 0.12, 7.55, and 11.82 forr BVB, LFT, and CKT, respecrespe tively. Note that the BVB model is very faithful to be a good alternative of the measured BRDF data for rendering. Figure 11 shows the results of B-spline spline volume representation repr sentation of the measured BRDF data of gold metallic paint where whe Bspline order is specified as p = q = r = 2,3,4 , and the number of control points is given in the form of show the average fitting errors AE. Figure 11 (b)) shows image difference error (nu + 1, nv + 1, nw + 1) = ( n, n, 4n) . Figures 11(a) shows (MAE) of the happy ppy Buddha images rendered by using the resulting BVB models.

(a)

(b)

(c)

(d)

(e) (f) (g) Figure 10.. Happy Buddha images rendered using BRDF models for gold metallic paint and their difference images. images

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

(a)

(b) Figure 11. Results of B-spline volume representation of the measured BRDF data of gold metallic paint: (a) plot of average fitting errors (AE); (b) plot of image difference errors (MAE). Note that, in Figure 11, the errors AE and MAE usually decrease as the number of control points increases. This trend is similarly noticed in BVB model-based fitting and rendering for other materials. Obviously, the quality of fitted BVB models is positively correlated with the quality of images rendered using the BVB models. This implies that in order to make BVB models appropriate for better quality of rendering, we need to reduce their fitting error as much as possible. However, there exists a trade-off between the fitting error and the number of control points of BVB models, so it is necessary to find a proper compromise between the fitting accuracy and the compactness of the BVB models. In addition, it is found from our extensive experiments that, for the same number of control points and the same B-spline order, the fitted BVB models of highly specular materials are likely to have larger fitting and rendering errors than those of the other materials. This means that the BRDF data of highly specular materials require more control points of BVB models to keep the same quality of fitting or rendering. Specifying the tolerance is much easier to control the quality of BVB models than specifying the number ( nu , nv , nw ) of control points since it is not known in advance how many control points are required. As an example of BVB model generation by specifying the tolerance (i.e. the error-bounded BVB model fitting), we constructed from the BRDF data of specular orange phenolic (AR=0.133, MR=2357.369) a set of B-spline volumes with different combinations of B-spline order and tolerance: B-spline order is specified as p = q = r = 2,3,4 , and the tolerance is given as rate = 0.50, 0.10, 0.05, 0.01, 0.005 . In this work, the tolerance is specified as tol = MR × rate where 0 < rate < 1 . Note that specular orange phenolic is highly specular (its MR value is ranked 22th among 100). Figure 12 shows results of error-bounded B-spline volume representation of the measured BRDF data of specular orange phenolic. Figure 12(a) shows the required number of control points of BVB models. Figure 12(b) shows image difference error (MAE) occurring in sphere images rendered using the BVB models. There exists a trade-off between the tolerance and the number of control points (correspondingly, the rendering quality). As the tolerance decreases, the fitted BVB models require more control points while the quality of rendering usually increases. This trade-off is similarly noticed in BVB model-based fitting and rendering for other materials.

(a)

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

(b) Figure 12. Results of error-bounded B-spline spline volume representation r presentation of the measured BRDF data of specular orange phenolic: (a) the number of control points required; (b) image difference error (MAE) of sphere images rendered dered using BVB models. Figure 13 shows the trade-off off more clearly. Figure 13(a) 13 shows the target image rendered using the TBF model. Figures 1 13(b) (b) to (d) show three test images rendered using BVB models m which are obtained with p = q = r = 2 and rate = 101 , 201 , 200 . The required numbers of control trol points in Figures 13(b) 13 to (d) are (49,58,8) , (80,83,10) , and (90,90,12) , respectively. Note that the BVB model obtained tained with the smaller tolerance represents the BRDF data more faithfully in rendering, but requires more control points. bounded BVB model fitting, we can find BVB models such that the rendering using them produces prod visualUsing the error-bounded ly plausible results. This task requires an iterative process repeating the following steps: error-bounded error bounded BVB model fitting, BVB-based based rendering, checking rendered images, and adjusting tolerances. From our practical experience, exp rience, it is suggested to specify the tolerance rate as follows: 0.3 ≤ rate ≤ 0.05 for diffuse materials ( MR < 50 ), and 0.05 ≤ rate ≤ 0.0005 for specular materials ( MR ≥ 50 ). It has been found by experience that low degree d B-splines splines are effective in BVB model fitting to BRDF data for specular materials, and high degree for diffuse materials.

(a)

(b)

Figure 14. Data sizee ratio required for BVB model fitting with visually plausible rendering results.

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

(c) (d) Figure 13. Sphere images rendered using BRDF models for specular orange phenolic: phenolic: (a) TBF; (b) BVB model ( rate = 0.1 , (nu , nv , nw ) = (49,58,8) ); (c) BVB model ( rate = 0.05 , (nu , nv , nw ) = (80,83,10) ); (d) BVB model ( rate = 0.005 , (nu , nv , nw ) = (90,90,12) ). For all the 100 materials, wee applied the iterative process to generate BVB models providing visually plausible plau rendering results, and investigated how many control points of the BVB models are required to represent the measured measured BRDF data sets. Figure 14 shows the data size ratios of the resulting BVB models, and the tolerance rates with which the BVB models are fitted to the measured BRDF data. The data size ratio means the ratio of the number of control points of a resulting BVB model to the number of BRDF data points. The average ratio ratio is 2.4%. The minimum ratio is 0.02% for blue-fabric blue with chrome with ( nu , nv , nw ) = (90,90, 26) . In the figure, BRDF ( nu , nv , nw ) = (5,5,13) , and the maximum ratio is 7.5% for chrome-steel materials were sorted in increasing order of their MR values. For most materials which are not highly specular, spec very low data size ratios (less than 1%) occur in BVB model generation. Even for highly specular materials, moderate data reduction ratios (2.5 to 7.5%) occur in BVB model generation. generation From these results, we found that BVB models are quite suitable for representing the measured BRDF data since they are compact in model size and effective in preserving the features of material appearance. Note that 165 megabytes meg of memory is required to store all the 100 BVB models, els, whereas 7 gigabytes of memory is needed to store all the 100 BRDF data sets in the form of TBF. For a scene where a significant number of different BRDF data sets are used, it is very difficult to perform TBF-based based rendering using a contemporary IBM-compatible IBM compatible personal computer because of excessive consumption of ranra dom access memory ory for the BRDF data sets, but it is easy to do with wit BVB-based rendering. Figure 15 shows an image of a simple scene rendered using BVB models where 100 spheres of different materials are placed.

Figure 155. Image of 100 spheres of different materials.

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

5.2 Comparison of BVB models with analytical models We compared BVB models with two kinds of analytical BRDF models (CKT and LFT) in accuracy of data fitting and in quality of rendering. Recall that we used the results of Ngan et al. [12] to fit the analytical models to the measured BRDF data sets provided by Matusik et al. [7,8]. As the materials were named slightly differently by the two research groups, we used 86 identically named materials for comparison among BRDF models. For BVB models to be compared with CKT and LFT models, we employed the BVB models which are fitted with the tolerance rates shown in Figure 14. Figure 16 shows the fitting errors AE of BRDF models (CKT, LFT, BVB) obtained by model fitting to the measured BRDF data of the 86 materials. In the figure, the errors were plotted in log scale and sorted in increasing order of MR value. Following the work of Ngan et al. [12], we omitted the BRDF values with incoming or outgoing angles larger than 80 to avoid possible outliers when computing the errors. Some BRDF models like CKT have extremely large errors when all the BRDF values are included, but the BVB models do not. Comparing the BRDF models in accuracy of data fitting, we found that the BVB models have much smaller fitting errors than the analytical models CKT and LFT. This becomes more obvious when the sectional views of iso-meshes of the BRDF models are displayed. Figure 17, for example, shows the sectional views ( φd = 180 ) of iso-meshes ( θi = 47 ) of three BRDF models generated from the measured BRDF data of nylon ( AR = 0.190 , MR = 796.479 ). Nylon is not specular, but special in that it has high peak values at graze angles. In the figure, cubic root was applied to scale down the data values. The BVB model fits the BRDF data very well while representing the overall features of the data. However, the CKT and LFT models have difficulty in representing the BRDF data.

0000

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

Figure 18 shows the plot of image difference errors (MAE) ( ) occurring in happy Buddha and sphere images rendered re using BRDF models (TBF, BVB, LFT, CKT) for the 86 materials. In the figures, the errors were plotted in log scale and sorted in increasing order of MR value. Comparing ing the BRDF models in quality of rendering, we found that the BVB models have much smaller image difference errors than the CKT and LFT models. This becomes more obvious by referring to Figures 68, and 10 that show the images rendered using four kinds of BRDF models (TBF, BVB, LFT, CKT). The images rendered using BVB models els are much more faithful to the target images rendered using TBF models (i.e., ( , measured BRDF data), whereas the images rendered using LFT and CKT models are not in most cases.

Figure 16. Comparison of BRDF models CKT, LFT, BVB in average fitting errors.

(a)) (b) (c) Figure 17. Comparison of sectional views of iso-meshes iso of fitted BRDF models ( θi = 47 , cubic root applied, nylon): (a) CKT; (b) LFT; (c) BVB.

(a)

(b) Figure 18. Comparison of image difference errors (MAE) occurring in physically-based physically based rendering using three BRDF models CKT, LFT, BVB: (a) happy Buddha images;; (b) sphere images.

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

From all the results mentioned above, we found that BVB models outperform the well-known analytical BRDF models (LFT and CKT) both in accuracy of data fitting and in quality of rendering.

6. Concluding Remarks In this paper, as one group of advocates asserting that B-spline models can be well suited for good representations of scientific data arising in computer graphics applications, we have proposed an adaptive method for B-spline volume representation (called BVB model) of BRDF data, and described the application of BVB models to photorealistic rendering. The proposed method basically performs approximate B-spline volume lofting, which decomposes the problem into three subproblems of multiple B-spline curve fitting along u-, v-, and w- parametric directions. Especially, it makes the efficient use of knots in the multiple B-spline curve fitting and accomplishes adaptive knot placement along each parametric direction of a resulting B-spline volume. We have demonstrated with extensive experiments that BRDF data coming from a wide range of materials can be represented by BVB models which are compact in model size and faithful in preserving the features of material appearance. In the Monte Carlo integration method [2], we need to provide a probability density function of BRDF for multiple importance sampling in ray tracing or incremental path generation of path tracing, which would further enhance the quality of rendered image with the same amount of computation. As a few analytical BRDF models including Phong [3] and Lafortune [4] support easy importance sampling, a reasonable approach to importance sampling for measured BRDF data has been to approximate the BRDF data by one of such analytical BRDF models. However, this also confronts the shortcomings of solving nonlinear optimization problems, and requires additional computation to evaluate the analytic form of a probability density function. In case of using TBF directly, a probability density function can be stored as another tabular form. However, this consumes a quite amount of storage size [2]. It is notable that Lawrence et al. [16] properly exploited a model based on factorization of BRDF data which is convenient for importance sampling. In a similar way of representing measured BRDF data in the form of tensor product B-splines, it is possible to approximate or represent the probability density function using the same form of B-splines. As we have implemented single-threaded programs to perform rendering, we found that the BVB model demands more computation for rendering than the others. For example, the average time to render happy Buddha images of 256x512 pixels is found to be 31, 34, 37, 50 minutes for LFT, CKT, TBF, BVB, respectively. The CKT model demands more computation than the LFT model. Although TBF-based rendering does not need any complicated computation, it takes more time than LFT or CKT-based rendering since it is a great computational burden to load BRDF data into memory and to read the data during a rendering process. BVB-based rendering usually demands more computation in evaluating model values using Equation (4) than the others. Thus, it takes more rendering time although using BVB models of lower degree somewhat reduces the rendering time. However, BVB models support the computation of BRDF values by matrix multiplication, but analytical models do not allow it since they are defined by non-polynomial equations. This means that the computation of BRDF values with BVB models can be much enhanced by adopting, for example, single instruction multiple data (SIMD) parallel computing [27]. We expect to do more research on the following topics for future works. First, as BVB models are not good in computation to render with single thread programs, we plan to implement advanced SIMD parallel computing to enhance the computation of BRDF values with BVB models and correspondingly the overall computational performance of BVB-based rendering. Second, we will devise a method for representing the probability density function of BRDF in the form of B-splines. Third, in order to deal with 4D BRDF data which come from anisotropic materials, we will extend the proposed method to generate a multivariate B-spline model from multivariate vector-valued scientific data.

References [1] Dorsey J, Rushmeier H, Sillion F. Digital modeling of material appearances. 1st ed. New York: Morgan Kaufmann; 2008. [2] Pharr M, Humphreys G. Physically Based Rendering: From Theory to Implementation, 1st ed. New York: Morgan Kaufman; 2004. [3] Phong BT. Illumination for computer generated pictures. Communications of the ACM. 1975; 18(6): 311-317. [4] Lafortune EPF, Foo SC, Torrance KE, Greenberg DP. Non-linear approximation of reflectance functions. In: Proceedings of ACM SIGGRAPH Computer Graphics; 1997. [5] Cook RL, Torrance KE. A reflectance model for computer graphics. ACM Transactions on Graphics. 1982; 1(1): 7-24. [6] Rusinkiewicz S. A survey of BRDF representation for computer graphics. Technical report: Stanford University; 1997. http://www.cs.princeton.edu/~smr/cs348c97/surveypaper.html. [7] Matusik W, Pfister H, Brand M, McMillan L. Efficient isotropic BRDF measurement. In: Proceedings of 14th Eurographics Workshop on Rendering; 2003. [8] Matusik W, Pfister H, Brand M, McMillan L. A data-driven reflectance model. ACM Transactions on Graphics. 2003; 22(3): 759– 769. [9] Optical Gauging Mechanism (OGM). http://www.dressingsim.com/DFL_en/product/OGM/. [10] Gunther J, Chen T, Goesele M, Wald I, Seidel HP. Efficient acquisition and realistic rendering of car paint. In: Proceedings of 10th

H. Park and J.H. Lee / Journal of Computational Design and Engineering 00 (2014) 0000~0000

0000

International Fall Workshop on Vision, Modeling, and Visualization; 2005. [11] Borshukov G. Measured BRDF in film production: realistic cloth appearance for “The Matrix Reloaded”. In: Proceedings of ACM SIGGRAPH Sketches & Applications; 2003. [12] Ngan A, Durand F, Matusik W. Experimental analysis of BRDF models. In: Proceedings of Eurographics Symposium on Rendering; 2005. [13] Schroder P, Sweldens W. Spherical wavelets: efficiently representing functions on the sphere. In: Proceedings of ACM SIGGRAPH Computer Graphics; 1995. [14] Koenderink JJ,van Doorn AJ, Stavridi M. Bidirectional reflection distribution function expressed in terms of surface scattering modes. In: Proceedings of the Fourth European Conference on Computer Vision; 1996. [15] Ozturk A, Kurt M, Bilgili A, Gungor C. Linear approximation of bidirectional reflectance distribution functions. Computers & Graphics. 2008; 32(2): 149-158. [16] Lawrence J, Rusinkiewicz S, Ramamoorthi R. Efficient BRDF importance sampling using a factored representation. ACM Transactions on Graphics. 2004; 23(3): 496-505. [17] Kurt M, Cinsdikici MG. Representing BRDFs using SOMs and MANs. SIGGRAPH Computer Graphics. 2008; 42(3): 1-18. [18] Hoschek J, Lasser D. Fundamentals of Computer Aided Geometric Design. 1st ed. Massachusetts: A K Peters; 1993. [19] Piegl L, Tiller W. The NURBS Book. 2nd ed. New York: Springer; 1995. [20] Lee JH, Park H. B-spline representation of measured BRDF for photorealistic rendering. In: Proceedings of the 3rd Korea-China Joint Conference on Geometric and Visual Computing (KCJC); 2007. [21] Lee JH, Park H. B-spline volume BRDF representation and application in physically-based rendering. Transactions of the Society of CAD/CAM Engineers. 2008; 13(6): 469-477. [22] Lee JH, Park H. Note on B-spline Volume Representation of Measured BRDF Data. In: Proceedings of 2010 Asian Conference on Design & Digital Engineering (ACDDE); 2010. [23] Park H, Kim K, Lee SC. A method for approximate NURBS curve compatibility based on multiple curve refitting. ComputerAided Design. 2000; 32(4): 237-252. [24] Park H, An approximate lofting approach for B-spline surface fitting to functional surfaces. International Journal of Advanced Manufacturing Technology. 2001; 18(7): 474-482. [25] Piegl L, Tiller W. Surface approximation to scanned data. The Visual Computer. 2000; 16(7): 386-395. [26] Park H, Lee JH. B-spline curve fitting based on adaptive curve refinement using dominant points. Computer-Aided Design. 2007; 39(6): 439-451. [27] Kurzak J, Alvaro W, Dongarra J. Optimizing matrix multiplication for a short-vector SIMD architecture - CELL processor. Parallel Computing. 2009; 35(3): 138-150.